160 XIII INTEGRATION
the set of all real-valued ^-measurable functions / such that/2 is integrable
is a vector space g&X, u) (also denoted by «£?£(//) or && For if / and g
are measurable and/2 and g2 are integrable, then/+# is measurable by
virtue of (13.11.2.1), and it follows from (13.9.13) that (/ + g)2 is integrable.
There exist functions/which are non-measurable, such that/2 is integrable
(Problem 2).
By abuse of language, the space S£ R(X, u) is called the space of square-
integrable functions. Since Np(af) = \a\Np(f) for p=l,2 and any scalar
a 7* 0, it follows from (13.11.2.1) that Np is a seminorm on the space
J$?£(X, u). The set Jf of functions / such that N//) = 0 is in both cases the
vector subspace of negligible (finite) real- valued functions (13.6.3). Hence
the space JS?£(X, jj) is not in general Hausdorff with respect to the topology
defined by the seminorm Np . The quotient space L£(X, ju) = J?fi(X, ju)/N
(also denoted by LJGu) or Lg) is the space of equivalence classes / of integrable
functions when p = 1, square-integrable functions when /? = 2 (13.6). A
function which is defined and finite almost everywhere in X is said to be
square-integrable if its class belongs to LR .
The number N//) is the same for all functions / belonging to a class
/eL£, and is denoted by Np(/). From (12.14.8) it follows that /H-»NP(/)
is a norm on L£ (p — 1 , 2).
(13.11.4) (Fischer-Riesz Theorem) The normed space LJ(X, u) is com-
plete (p = 1, 2) (in other words, it is a Banach space). More precisely:
(i) .//* (gn) is a sequence of functions on X whose classes belong to Lg,
oo
< + oo, £/ze« ?//e jenas whose general term is gn(x) converges
absolutely in R almost everywhere. If g(x) = £ gn(x), the class g of the func-
n~ 1
oo
tion g (defined almost everywhere) belongs to Lg , and we have g = £ gn
n= 1
in the normed space LJ .
(ii) If (/„) is a sequence of functions such that the sequence (/„) of their
equivalence classes is a Cauchy sequence in L£ , then there exists a subsequence
(/nj such that (fnk(x)) converges to a limit f(x) in R almost everywhere. For
each such subsequence (fn]), the class ? of f belongs to L£ and is the limit of
the sequence (/„) in the normed space Lg .
(iii) Let (/„) be a sequence of functions belonging to JS?g(X, u) such that
the sequence (fn(x)) converges almost everywhere to a limit /(x), and suppose
that there exists a function h e ^g(X, u) such that \fn(x)\ g h(x) almost every-
where, for all n. Then the class / off belongs to L£ and is the limit of the
sequence (/„) in the normed space L£ .Use (a) to prove that the condition is necessary.)